On Stability of a Nonlinear, Periodically Time Varying, Rotating Blade

2011 ◽  
Author(s):  
Fengxia Wang
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Multiple Time ◽  
Time Varying ◽  
Rotating Blade ◽  
Stability And Bifurcation ◽  
Geometric Stiffening ◽  
Rotational Motions ◽  
The Stability

This paper discusses the stability of a periodically time-varying, spinning blade with cubic geometric nonlinearity. The modal reduction method is adopted to simplify the nonlinear partial differential equations to the ordinary differential equations, and the geometric stiffening is approximated by the axial inertia membrane force. The method of multiple time scale is employed to study the steady state motions, the corresponding stability and bifurcation for such a periodically time-varying rotating blade. The backbone curves for steady-state motions are achieved, and the parameter map for stability and bifurcation is developed. Illustration of the steady-state motions is presented for an understanding of rotational motions of the rotating blade.

Nonlinear Oscillation of a Rotor-AMB System With Time Varying Stiffness and Multi-External Excitations

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
M. Kamel ◽  
H. S. Bauomy
Keyword(s):  
Steady State ◽  
Order Approximation ◽  
Nonlinear Differential Equations ◽  
Steady State Solution ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Multiple Time ◽  
Time Varying ◽  
The Stability ◽  
Amb System

The rotor-active magnetic bearing system subjected to a periodically time-varying stiffness having quadratic and cubic nonlinearities is studied and solved. The multiple time scale technique is applied to solve the nonlinear differential equations governing the system up to the second order approximation. All possible resonance cases are deduced at this approximation and some of them are confirmed by applying the Rung–Kutta method. The main attention is focused on the stability of the steady-state solution near the simultaneous principal resonance and the effects of different parameters on the steady-state response. A comparison is made with the available published work.

scholarly journals Projective Synchronization Analysis of Drive-Response Coupled Dynamical Network with Multiple Time-Varying Delays via Impulsive Control

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Song Zheng
Keyword(s):  
Impulsive Control ◽  
Sufficient Conditions ◽  
Projective Synchronization ◽  
Multiple Time ◽  
Time Varying ◽  
Comparison Theory ◽  
Dynamical Network ◽  
Functional Differential ◽  
The Stability ◽  
System Nodes

The problem of projective synchronization of drive-response coupled dynamical network with delayed system nodes and multiple coupling time-varying delays is investigated. Some sufficient conditions are derived to ensure projective synchronization of drive-response coupled network under the impulsive controller by utilizing the stability analysis of the impulsive functional differential equation and comparison theory. Numerical simulations on coupled time delay Lorenz chaotic systems are exploited finally to illustrate the effectiveness of the obtained results.

Frequency response analysis of periodically time varying circuits

2018 ◽  
Vol 37 (3) ◽  
pp. 1204-1213
Author(s):  
Igor Korotyeyev
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Steady State Solution ◽  
Variable Coefficients ◽  
State Solution ◽  
Response Analysis ◽  
Time Varying ◽  
Content Type ◽  
Frequency Responses ◽  
Definition Of

Purpose The purpose of this paper is to introduce a method for the analysis of steady-state processes in periodically time varying circuits. The method is based on a new definition of frequency responses for periodic time-varying circuits. Design/methodology/approach Processes in inverter circuits are often described by differential equations with periodically variable coefficients and forcing functions. To obtain a steady-state periodic solution, the expansion of differential equations into a domain of two independent variables of time is made. To obtain differential equations with constant coefficients the Lyapunov transformation is applied. The two-dimensional Laplace transform is used to find a steady-state solution. The steady-state solution is obtained in the form of the double Fourier series. The transfer function and frequency responses for the inverter circuit are introduced. Findings A set of frequency characteristics are defined. An example of a boost inverter is considered, and a set of frequency responses for voltage and current are presented. These responses show a resonance that is missed if the averaged state-space method is used. Originality/value A new definition of frequency responses is presented. On the basis of frequency responses, a modulation strategy and filters can be chosen to improve currents and voltages.

scholarly journals Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1078 ◽  
Author(s):  
Mohamad Mustaqim Junoh ◽  
Fadzilah Md Ali ◽  
Ioan Pop
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Stagnation Point ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Governing Equations ◽  
Induced Magnetic Field ◽  
The Stability ◽  
Stretching Or Shrinking Sheet

The revised Buongiorno’s nanofluid model with the effect of induced magnetic field on steady magnetohydrodynamics (MHD) stagnation-point flow of nanofluid over a stretching or shrinking sheet is investigated. The effects of zero mass flux and suction are taken into account. A similarity transformation with symmetry variables are introduced in order to alter from the governing nonlinear partial differential equations into a nonlinear ordinary differential equations. These governing equations are numerically solved using the bvp4c function in Matlab solver, a very adequate finite difference method. The influences of considered parameters ( P r , M, χ , L e , N b , N t , S, and λ ) on velocity, induced magnetic, temperature, and concentration profiles together with the reduced skin friction and heat transfer rate are discussed. Results from these criterion exposed the existence of dual solutions when magnetic field and suction are applied for a specific range of λ . The stability of the solutions obtained is carried out by performing a stability analysis.

Steady-state analysis of DC converter using Galerkin’s method

2019 ◽  
Vol 38 (6) ◽  
pp. 2057-2069
Author(s):  
Igor Korotyeyev
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Galerkin Method ◽  
State Space ◽  
Time Interval ◽  
Time Varying ◽  
Content Type ◽  
State Process ◽  
The Galerkin Method ◽  
Steady State Process

Purpose The purpose of this paper is to present the Galerkin method for analysis of steady-state processes in periodically time-varying circuits. Design/methodology/approach A converter circuit working on a time-varying load is often controlled by different signals. In the case of incommensurable frequencies, one can find a steady-state process only via calculation of a transient process. As the obtained results will not be periodical, one must repeat this procedure to calculate the steady-state process on a different time interval. The proposed methodology is based on the expansion of ordinary differential equations with one time variable into a domain of two independent variables of time. In this case, the steady-state process will be periodical. This process is calculated by the use of the Galerkin method with bases and weight functions in the form of the double Fourier series. Findings Expansion of differential equations and use of the Galerkin method enable discovery of the steady-state processes in converter circuits. Steady-state processes in the circuits of buck and boost converters are calculated and results are compared with numerical and generalized state-space averaging methods. Originality/value The Galerkin method is used to find a steady-state process in a converter circuit with a time-varying load. Processes in such a load depend on two incommensurable signals. The state-space averaging method is generalized for extended differential equations. A balance of active power for extended equations is shown.

scholarly journals On Multi-Laplace Transform for Solving Nonlinear Partial Differential Equations with Mixed Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana ◽  
Suares Clovis Oukouomi Noutchie
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Nonlinear Partial Differential Equations ◽  
Order Derivative ◽  
Partial Differential ◽  
Analytical Technique ◽  
Novel Approach ◽  
Mixed Derivatives ◽  
The Stability

A novel approach is proposed to deal with a class of nonlinear partial equations including integer and noninteger order derivative. This class of equations cannot be handled with any other commonly used analytical technique. The proposed method is based on the multi-Laplace transform. We solved as an example some complicated equations. Three illustrative examples are presented to confirm the applicability of the proposed method. We have presented in detail the stability, the convergence and the uniqueness analysis of some examples.

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Erik Sweet ◽  
Kuppalapalle Vajravelu ◽  
Robert Gorder
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Polar Coordinates ◽  
Rotating Flow ◽  
Rotating Sphere

AbstractIn this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.

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